#117520

order	512
nilpotency class	3
isoclinism	isoclinic class of #10176
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,6)(3,4)(7,8)(9,10,13,18,14,17,23,16)(11,19,15,12,20,21,24,22);(2,7)(3,4)(6,8)(9,11,13,15,14,20,23,24)(10,12,16,19,17,22,18,21);(1,2,5,8)(3,7,4,6)(9,12,14,22)(10,15,17,24)(11,18,20,16)(13,21,23,19)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=i,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24716);